Technique for measuring the complex elastic (young{3 s) modulus utilizing laser interferometry

ABSTRACT

The dynamic complex elastic modulus of a viscoelastic material is computed from three longitudinal deflection measurements on a prismatic rod of the material which is subjected to a compressional sinusoidal stress wave. A laser interferometer is proposed to make these deflection measurements in combination with reflecting mirrors which are located at the top, half and quarter positions of the sample rod. The rod is secured at one end and free at the distal end where the compressional stress wave is applied.

1 CR5 3; 90 1 i 074/ Q Ma United Statw t a... 1111 3,901,074

Douglas Aug. 26, 1975 TECHNIQUE FOR MEASURING THE 3,699,808 10 1972 Fordet a1, 73 91 COMPLEX ELASTIC (YOUNGS) MODULUS g e c vlg e a. UTILIZINGLASER INTERFEROMETRY 3,782,184 l/197 4 Shuck 73/101 Inventor: Bruce E.Douglas, Edgewater, Md.

The United States of America as represented by the Secretary of theNavy, Washington, D.C.

Filed: Feb. 11, 1974 Appl. No.: 441,536

[73] Assignee:

References Cited UNITED STATES PATENTS 5/1967 Barigant ..73/67.2 12/1968Barigant ..73/67.1

NARROW BAND 4 ANALYZER X-Y PLOTTER SCOPE /2O Primary ExaminerRichard C.Queisser Assistant Examiner.l0hn P. Beauchamp Attorney, Agent, or FirmR.S. Sciascia; Q. E. Hodges 5 7 ABSTRACT The dynamic complex elasticmodulus of a viscoelastic material is computed from three longitudinaldeflection measurements on a prismatic rod of the material which issubjected to a compressional sinusoidal stress wave. A laserinterferometer .is proposed to make these deflection measurements incombination with reflecting mirrors which are located at the top, halfand quarter positions of the sample rod. The rod is secured at one endand free at the distal end where the compressional s t r :ss wave isapplied.

2 Claims, 2 Drawing Figures MIRROR VIBRATION GENERATOR Pmminwsz m 901 m4SHZU 1 BF 2 [22 l| X-Y NARROW BAND PLOTTER ANALYZER VIBRATION GENERATORl5 FIG.

PAIENIEII Auszs I975 SEEZEI 2 [IF 2 I H 4 m 1 I0, I III I I. I x n. I II. l 2 c 4 I w .m .l I I s I D. W 8 Il H I O 2 e 4 I I O O h R O I I O Om II I o z o I I I 0 O 6 n I I I E d O! L I 2 2 2 2 2 2 2 2 l l l I I lI l l I O O O O O O O O O O IOOO I FREQUENCY, HZ lR l AND|R SPECTRA FORA VISCOELASTIC ROD TECHNIQUE FOR MEASURING THE COMPLEX ELASTIC (YOUNGS)MODULUS UTILIZING LASER INTERFEROMETRY The invention described hereinmay be manufactured and used by and for the Government of the UnitedStates of America for governmental purposes without the payment of anyroyalties thereon or therefor.

BACKGROUND AND SUMMARY Viscoelastic materials are commercially utilizedto damp mechanical vibrations in resonant structures. In general, thedesign of a damping treatment which utilizes these materials requiresthat two of the materials dynamic complex moduli be known as a functionof frequency and temperature. The invention described herein is a methodand apparatus of measuring one of these moduli, the complex elastic, orYoungs, modules, which utilizes a laser interferometer to measure thelongitudinal vibrational displacement at three positions on a sample ofthe material in the form ofa viscoelastic prismatic rod which issubjected to a compressional sinusoidal stress load. The rod is fixed atone end and free at the other. Although several techniques exist forthis type of measurement, the instant method is effective over a widerange of enviromental conditions, e.g., a wide range of temperaturesunder which the parameter can be determined. It offers a precise,nonresonant method for determining the complex elastic modulus ofinherently damped materials which have a high real elastic modulus.Experimentally, this procedure places a negligible mechanical load onthe test sample to minimize the effects of external mechanical loadingon the test sample.

BRIEF DESCRIPTION OF DRAWINGS FIG. 1 is a schematic view of theapparatus used in this invention.

FIG. 2 is a graph of frequency versus l R l and lR.,l the displacementratios for an example material.

DETAILED DESCRIPTION Referring to FIG. 1, laser 1 produces a coherentcollimated beam of light 16 which is split at a beam splitter 2, thefirst beam 16a is reflected at a fixed mirror 3 back to the beamsplitter where it recombines with the second beam 16!; which isreflected from one of the mirrors 5", or 5" which are attached to theviscoelastic sample. A mirror 4 is fixed against vibration, but slidablealong base 17 to be positioned under mirror 5, 5", or 5 and redirectsthe second beam so that the mirrors at 5, 5", 5 can be attached to therod 7 to measure longitudinal displacements. The second beam 16 b isretroreflected from one of the mirrors 5, 5", or 5" back along the samepath 16b. The two beams 16a and 1612 then recombine at the beam splitter2 where they produce interference fringes which are detected by aphotodiode 9. It is to be understood that FIG. 1 shows for convenienceonly one example where the mirror 4 is under only one of the mirrors 5',but is slidable to positions under mirrors 5" and 5". A vibrationgenerator, e.g., a piezoelectric crystal or magnet driven by a coil, ismechanically attached to the viscoelastic prismatic rod 7 and driven byan oscillator 14 and power amplifier 15 to produce a compressionalsinusoidal stress wave at the top of the rod 7. The rod is enclosed byan environmental control chamber 8. The

three mirrors 5', 5", and 5" are attached to the rod having a length L,at the free end, i.e. at L, the half length position, L/2, and thequarter length position, L/4 from the fixed point of attachment of therod, indicated at 17 in FIG. 1. The mirrors are bonded to the rod sothat they are displaced as the point of attachment of the individualmirrors is displaced by the sinusoidal loading. A retroreflected beam oflight from the motion of the mirrors 5, 5", 5" produce lightinterference fringes at the photodiode 9, after transmission through aconventional iris l8, and are related to the displacement of the mirrorsduring vibrational displacement. The interference fringes measured bythe photodiode 9 is a measure of the longitudinal vibration deflection.The longitudinal displacement of the the sample rod 7 is designated U Uand U for the free end, half length and quarter length positions,respectively, in the equations set forth hereinafter.

The photodiode 9 electronically counts the fringes which givesa measureof the displacement or me attached mirrors 5. The time varyingphotodiode voltage is then amplified by amplifier 10, observed foradjustment on an oscilloscope 20, and filtered by filter 11 at thefrequency of excitation of the rod or one of its harmonics (depending onthe method used i.e. either the reference or ratio method to obtaindisplacement readings). Any of the standard techniques such as an X-Yplotter 22 for displaying and for calculating displacement from asuitably analyzed photodiode signal can then be employed to obtain thedisplacement of the mirrors 5, 5", and 5" which corresponds to thelongitudinal displacement of the rod at the point of attachment. Exceptfor the mirror attachments on the viscoelastic sample the abovedescripiton is just that of a standard laser interferometer. The pointsof attachment of the mirrors 5', 5", 5" on the sample rod 7 are criticalto permit the use of the displacement measurements to determine thedynamic complex elastic modulus E*.

Having obtained this set of three longitudinal deflection measurements UU U by the standard methods described above, these readings can then besubstituted into the'expressions mzi I L/Zl I to obtain IRA and Rd atthe driving frequency,f, of

the vibration generator. These values for R and R I can then besubstituted into the following expressions frequency which the rod wasdriven and at the temperature under which the measurements were made.

A plot of frequency versus lR- l and |R is made from the displacementmeasurements, and an example of such a spectrum is shown in FIG. 2.

For frequencies up to the last maximum of |R spectrum the dynamiccomplex elastic modulus can be determined by:

since [3* E(l +ia') and where:

p mass density of the rod;

f= the frequency, in Hertz; and,

n an integer whose absolute value is equal to the number of maxima inthe lR spectrum below the driving frequencyfand whose sign is oppositethe sign of the slope of R at frequencyf.

For frequencies above the frequency of the last maximum of RA determinethe dynamic complex elastic modulus E* according to the followingrelationship:

(l-i-id) where,

d=tan2 [arcsin (i f ln where;

c the velocity of the longitudinal wave at frequency, f, measured bywell known laser interferometric techniques.

Some modifications and variations of the present invention are possiblein light of the above teachings. It is therefore to be understood thatthe invention may be practiced otherwise than as specifically described.

What is claimed is:

l. A method for measuring the dynamic complex elastic modulus of aviscoelastic material comprising the steps of:

fixedly mounting a prismatic rod of the material of length L at one ofits ends;

maintaining the rod at a predetermined temperature;

introducing a sinusoidal longitudinal stress wave of a selected numberof frequencies within a predeter mined range of frequencies into therod;

5 measuring the deflection, U of the rod at its free end, at a pointmidway its length U and at a point one quarter of the length of the rodfrom its fixed end. U for each of the selected frequen- CleS;

A. for frequencies below the last maximum in |R determine the complexelastic modulus, E* for the given frequencyfaccording to the followingrela tionship p mass density of the rod f= the frequency in Hertz; and

n an integer determined at the given frequency, f, such that theabsolute value ofn equals the number of peaks in |R for the |R Ispectrum below frequencyfand the sign ofn is opposite the sign of theslope of R for values offup to the last maximum 0 in R4 B. forfrequencies above the last maximum in |R determine the complex elasticmodulus E* according to the following relationship:

5* (hwmw (l +z'd) where;

d=tan {2 arcsin In R2 c the velocity of the longitudinal wave atfrequency 2. An apparatus for measuring the displacement of a sample rodof viscoelastic material under sinusoidal longitudinal compression wavesat lineal points critical for the determination of the dynamic complexmodulus of elasticity of the material, comprising:

an environmental housing for enclosing the sample rod and controllingthe temperature thereof;

means for securing one end of said rod in said environmental housing;

means attached to the unconstrained end of the sample rod forintroducing sinusoidal compression stress waves therein;

means for producing a beam of collimated light and introducing the sameinto said environmental housa point one-quarter the length from thesecured end;

means for combining said split beams reflected from said reflectingmeans with the other of said split beams; and

means for counting the interference fringes produced by the combiningmeans thus determining the displacement of each of said reflecting meansat the 3 selected frequency.

1. A method for measuring the dynamic complex elastic modulus of aviscoelastic material comprising the steps of: fixedly mounting aprismatic rod of the material of length L at one of its ends;maintaining the rod at a predetermined temperature; introducing asinusoidal longitudinal stress wave of a selected number of frequencieswithin a predetermined range of frequencies into the rod; measuring thedeflection, UL, of the rod at its free end, at a point midway its lengthUL/2, and at a point one quarter of the length of the rod from its fixedend, UL/4, for each of the selected frequencies; A. for frequenciesbelow the last maximum in R4 determine the complex elastic modulus, E*for the given frequency f according to the following relationship
 2. Anapparatus for measuring the displacement of a sample rod of viscoelasticmaterial under sinusoidal longitudinal compression waves at linealpoints critical for the determination of the dynamic complex modulus ofelasticity of the material, comprising: an environmental housing forenclosing the sample rod and controlling the temperature thereof; meansfor securing one end of said rod in said environmental housing; meansattached to the unconstrained end of the sample rod for introducingsinusoidal compression stress waves therein; means for producing a beamof collimated light and introducing the same into said environmentalhousing to impinge upon said sample rod; means for splitting said beamof collimated light into two beams, said means interposed along saidbeam of light; means for reflecting one of said split beams, mountedperpendicularly along the length of said sample rod, one forsuccessively measuring the longitudinal displacement of said sample rodalong the length thereof at the unconstrained end, another at a pointmidway along its length, and still another at a point one-quarter thelength from the secured end; means for combining said split beamsreflected from said reflecting means with the other of said split beams;and means for counting the interference fringes produced by thecombining means thus determining the displacement of each of saidreflecting means at the selected frequency.